Research construction: R420 target-basis compensation saves 7,879 XORs within the existing 192-bit nine-stage prefix charge (no new κ)

[commdata2338]

Change and mathematical scope

Status: research construction with exact verification. No new κ. The selected certificate, its κ (51110230636731/31250000000000000) and upstream/ are unchanged. All files are new, under research/r420-target-compensation/.

eumemic's #543 R420 bit word implements its frame-zero dirty-helper compensation D, y <- y + K s for a fixed binary 120-by-420 matrix K, with 19,922 XORs. This PR factors K through a reversible target basis. D first applies 205 target transvections y[a] ^= y[b], whose product is E. It then adds helpers along the 11,633 nonzero entries of R = E K, and finally repeats the 205 transvections in reverse order. Over F2 the result is E^-1 (E y + R s) = y + K s for arbitrary incoming targets and helpers. D never writes a helper or a source and needs no workspace.

Quantity	#543	This PR
D XORs (entrance + helper injections + exit)	19,922	12,043 (205 + 11,633 + 205)
Complete native word, XORs	26,916	19,037
Hosted instructions, 288 local calls	8,603,712	6,334,560
Forward / reverse positive bounds F / B	144,651 / 2,270,064	144,651 / 4,409,580
Local payload 64 F^3 B^2, bits (cap 104)	100	102
Nine-stage prefix 2^18 F^5 B^4 / charge, bits	189 / 192	192 / 192
Two-level coefficient / charge, bits	416 / 512	424 / 512

• Unchanged. Only D changes. The 420 helpers, the nine COPY/ERASE pairs, every register's frame path, the paid profile {1:1320, 2:1440, 4:60, 5:120, 8:9} with rank mass 5,112, and the recursive children stay as they are. The bit lower bound and the selected κ reproduce exactly.
• Prefix. The reverse bound grows. The search admitted a row step only if F stayed at most 144,651 and B at most 4,409,638, the largest B with 2^18 F^5 B^4 < 2^192, so the existing 192-bit nine-stage and 512-bit two-level charges still hold. The unrounded invoice coefficient increases. The claim is fewer instructions under unchanged charges, not a measured speedup.
• Changed hypothesis. The new D breaks Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543's gate-local H2, under which D gates read only helpers, and Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543's direct K_equals_D extraction no longer applies. The verifier records both predicates as false and keeps every other H1–H5 check. In their place, README.md proves a semantic D-boundary lemma: D runs at frame zero and implements exactly (x, y, s) -> (x, y + K s, s), so the unchanged remainder of the word sees identical values and frames. The verifier checks that boundary on all 660 symbolic columns.
• Search. OS seed 18 chose this object (18 mod 3 = 0 among three candidates). At each step a greedy pass scored every ordered pair of target rows, kept only steps that passed an exact incremental prefix check, and broke ties with the seed. It stopped after 205 steps and 2,941,680 pair evaluations, when no admissible positive-gain pair remained; the cap was 1,000 steps. An unrestricted control with the same seed reached 10,384 D XORs but needed 201 nine-stage bits and broke the 104-bit payload cap, so it is kept only as a rejected control. 12,043 is not claimed optimal.

Predecessor work. Details and pins for 39 source files are in README.md and SOURCES.json.

• eumemic's Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543, reviewed at 2065fe4f72a914d907f1046603f5a36f17018535 and building on Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493, supplies the word, the compensation matrix, the interleaved interface and the checker this package reuses.
• Credit carried from Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543's lineage: Dylan Crow (E8 word, record and checker lineage), DreamingOfClouds (Bit word R480: totals on dead registers, c = 0.001574753352953808650916 (+0.595% over #493) #503/Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510, generalized nine-stage accounting), DaysSky (bridged columns, whole-call hosting), Zhihao Chen (mutable-source/common-label and compressed geometry) and Chafik Boukhalfa (physical routes, templates and audits). Chafik Boukhalfa's selected Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626 supplies the complex lower bound that the hosted replay cites.
• Dugongue's Conditional kappa = 0.001291542455423938: two-layer target factoring on R715 (+0.1461% over #470) #476 and Conditional kappa = 0.001295508560071773: fourteen target factors on PR473 (+0.1226%) #478 used the reversible donor-receiver target identity on R715 phase B. Each transvection pair here is that identity, applied to the compensation.
• Mikhail R. Gadelha's concurrent Research: save 108 R420 XORs with restored frame-zero sharing (no new kappa) #686 published the semantic D-boundary substitution first. This PR claims no priority for it or for basis factoring.

Open obligations.

• A maintainer must accept the semantic D-boundary lemma in place of the gate-local H2 and update the supplier's source bindings.
• B sits 58 below the nine-stage cutoff, so combining this D with any change that raises F or B needs a fresh prefix recount.
• The all-size compiler, common-label/mutable-basis realization, pooled hosting, normalized residual, precision/tape recovery and analytic-transfer assumptions remain inherited.

Research note and proof, frozen recipe, certificate, hosted certificate, source pins, validation record, seed, search and overlap record, verifier and search.

Overlap. No coordination issue exists; the only open issue, #650, asks about Discussions. The circuit was nominated and frozen locally before any concurrent D PR was read (RESEARCH.json records the freeze at 04:19 UTC on 13 October), and nothing from those PRs is imported. The package README records the overlap as of 04:25 UTC, when only #686 and #694 had appeared. A refresh between 05:28 and 05:37 UTC searched PR titles and bodies through #788 and read every PR among them that changes D. main is still 66849e1, so nothing on D has merged. Many open PRs now change this D, and two reach lower counts:

PR	Basis and recipe class	D XORs	Native XORs	Nine-stage bits
This PR	target; 205 successive steps	12,043	19,037	192
#686	helper; one pair	19,814	26,808	189
#694	helper; 201 disjoint pairs	12,988	19,982	192
#713	target; 57 disjoint pairs	16,772	23,766	191
#716	target; 60 disjoint pairs, optimal among disjoint pairs	16,672	23,666	191
#742	target; shared donors	16,634	23,628	192
#759	target; rooted forest	15,138	22,132	197
#774	target; forest with optimized cuts	15,025	22,019	197
#733	#694's helper basis, then 57 target pairs	11,554	18,548	195
#681	#694's helper basis, then 56 target pairs, in a six-PR consolidation	11,562	18,378	196

The last column applies this package's nine-stage formula to each PR's reported F and B. The other D PRs check the local payload against 2^104 and leave the hosted prefix to integration; #681 reports 196 bits itself, and the formula reproduces it. #681's native count includes its phase-A, phase-B and decoder changes.

• Lower counts. Scott Woolsey Biggart's Research: compose R420 bases for 8,368 fewer XORs #733 and the D stage of Alluri's Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681 use 489 and 481 fewer D XORs than this PR. Both exceed the 192-bit nine-stage charge: Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681 raises its allocation to 256 bits and replays the hosted invoice at that charge, and Research: compose R420 bases for 8,368 fewer XORs #733 leaves global invoices open. This PR does not settle whether their extra savings are worth the larger allocation.
• What this adds. Of the D constructions that keep the existing 192-bit charge, this one saves the most, 945 XORs more than Greer Guthrie's Research: save 6,934 R420 XORs with a restored helper basis #694. Apart from Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681's consolidation, it is the only D package that replays the complete nine-stage hosted invoice. Its D of 12,043 lies below the 14,921-XOR floor that Oscar Fennell's Research: rooted R420 target forests save 4,784 XORs with exact class floor #759 certifies for rooted target forests, a floor Anthony's Research: certify R420 forest cuts under the 104-bit cap #774 corroborates, so this circuit lies outside that class, as it lies outside the disjoint pairs of Sagar Thakrar's Research: optimal disjoint R420 target pairs save 3,250 XORs (no new kappa) #716. Scott Woolsey Biggart's Research: save 3,150 R420 XORs with a restored target basis #713 published the first target-side circuit, at 04:40 UTC, followed by Research: optimal disjoint R420 target pairs save 3,250 XORs (no new kappa) #716 and ellipsi2's Research: shared target donors save 3,288 R420 XORs (no new kappa) #742.
• Composition and conflict. These PRs replace the same D subword, so they are alternatives; no file paths collide. Research: compose R420 bases for 8,368 fewer XORs #733 shows that a helper basis and a target basis compose. At Research: save 6,934 R420 XORs with a restored helper basis #694's F = 202,699, however, the 192-bit charge allows B only up to 2,892,279, against Research: save 6,934 R420 XORs with a restored helper basis #694's own 2,446,788, so this PR attempts no such composition. cxnmai's Research: decoder source cosets save 244 R420 XORs (no new kappa) #748 changes K itself through a decoder-total representation, which would require recomputing this recipe on the new matrix. Independent verification: R420 compensation rank and minimum-weight kernel #731 (Joshua Snider) and Independent verification: ten optimal tests for R420 compensation faults #762 (Sagar Thakrar) verify the baseline K and complement this work. Research: save 48,400 JP8 primitives with a restored compensation basis (no new kappa) #749 (Sricharan Vagvala) takes target pairs to the JP8 complex supplier. The phase-A, phase-B and cleanup PRs (Research: simplify one R420 block from 50 to 33 XOR gates (no new kappa) #670, Research: reduce R420 block 5 from 46 to 34 XOR gates (no new kappa) #674, Research: compose R420 circuits with 16,704 fewer hosted instructions (no new kappa) #688, Research: shorten R420 phase-A quad 13 from 19 to 12 XORs (no new kappa) #691, Research: save 114 R420 XORs by sharing restored decoder operands (no new kappa) #708, Research: shorten R420 cleanup by two XORs with certified conjugation (no new kappa) #710, Research: restored decoder stars save 128 R420 XORs (no new kappa) #753, Research construction: save 137 R420 cleanup XORs with exact four-wire synthesis #754) change other subwords; combining any of them with this D needs the prefix recount above.

Validation

From the repository root (Python 3.11+, standard library only):

python3 -I -B research/r420-target-compensation/verify.py

Output, re-run on base 66849e1 with this package before submission (Linux, Python 3.12.3, about 3 seconds):

PASS_R420_TARGET_COMPENSATION D=19922->12043 total=26916->19037 columns=660 rank_mass=5112 nine_bits=192 controls=13

python3 -I -B research/r420-target-compensation/search.py reruns both seed-18 trials and compares the full result with the frozen recipe. It printed PASS_REPRODUCIBLE_SEARCH guarded_steps=205 D_adds=12043 trials=2 seed=18 in the same pre-submission run.

The optional complete hosted replay needs a C++17 compiler, Boost headers, and requirements.txt (SymPy 1.14.0, mpmath 1.3.0) in a virtual environment. It writes a fresh directory under build/:

python3 -I -B research/r420-target-compensation/replay_hosted.py

Expected final status:

PASS_R420_TARGET_COMPENSATION_HOSTED local_adds=19037 hosted_events=6334560 prefix_charge=192 coefficient_charge=512 kappa=unchanged

• Finite checks.
  • verify.py runs Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543's exact lattice, frame and symbolic-row engine, plus a separate Boolean walk of all 660 columns in each direction. It checks the frozen recipe without trusting or rerunning the search, and rejects all 13 negative controls. One control omits the whole entrance basis: it passes when targets start at zero but fails the arbitrary-target boundary.
  • The hosted replay freshly compiles the predecessor C++ engines. It checks 900 five-stage and 1,380 nine-stage formal columns with 6 and 18 omitted-bridge controls, and 4,740 unbanked parent columns; the parent and hosted audits reject 17 and 16 mutations. Four record controls omit a basis ADD, a helper ADD or a COPY, or understate a MOVE rank. All 47 outer assembly inequalities reproduce the selected κ from Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626's pinned complex lower bound. The recorded run, on Linux with Python 3.12.3, SymPy 1.14.0, mpmath 1.3.0 and GCC 13.3.0, produced a full report equal to HOSTED-CERTIFICATE.json. It was not re-run for this submission.
• Proof arguments. README.md proves the D-boundary lemma, its substitution into the unchanged word, and the exact incremental formula the search uses to admit steps. None of these is inferred from the finite replay. The identity holds over F2; the positive integer bounds are recomputed from the literal instructions in both orientations.
• Repository checks. make selected-record-check passes: PASS selected record, exact kappa, original source inventory and proof supplement: 51110230636731/31250000000000000. git diff --check is clean, and no existing file or upstream/ byte changes. The verifier above was re-run on the submitted commit.
• Not claimed. This is not a formal verification of the multiplication theorem or a cold replay of the complex supplier; Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626's complex bound is pinned and cited. It claims no circuit optimality, no measured speedup and no composition with any other D, phase-A, phase-B or cleanup change. No repository-wide suite, eleven-stage physical replay, remote CI or Lean pass was run.

Sprint: bit-supplier track, OS seed 18. The claim and its result update are recorded locally. Submitted by @commdata2338 with substantial AI assistance, under Apache-2.0: OpenAI Codex (GPT-6-Astra) wrote the analysis, proof and checker, and Claude Code packaged the submission. No independent human peer review is claimed.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG

[Mister-Meeseeks]

100-PR batch review at 6be7364e0245c5a1c931cb1c43b3da991b755850.

This remains open after scoped triage. Next review step: Local certificate passes; review the additional hosted replay and cumulative claims before whole-head incorporation. Local semantic D validity alone is not hosted supplier admission.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.